Vol 9, No 3 (2018)

International Journal of Mathematical Archive-9(3), 2018,

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International Journal of Mathematical Archive- 9(3), March – 2018 165

CONFERENCE PAPER

National Conference dated 26th _{Feb. 2018, on “New Trends in Mathematical Modelling” (NTMM - 2018),}

Organized by Sri Sarada College for Women, Salem, Tamil Nadu, India.

IMPROVING THE CONDITION

OF TRANSPORTATION IN SALEM CITY USING GAME THEORETICAL APPORACH

B. AMUDHAMBIGAI

_{, A. NEERAJA}

_{AND R. ROSELEEN NITHIYA}

Department of Mathematics,

Sri Sarada College for Women, Salem-636016, Tamil Nadu, India.

E-mail: [email protected]1_{, [email protected]}2_{, [email protected]}3_.

ABSTRACT

W

Key Words: Transportation, matrix of losses and function of risk.

2010 Mathematics Subject Classification: 90C70, 90C26.

1. INTRODUCTION

When mathematical models are used to make decisions, certain preliminary activities are need to be done. First, one should determine the parameters which affect the final result. For every parameter one needs to define the value intervals which are of significance for the studied problem.

Transportation in a city is always the most important factor for the people. To serve the needs of increasing population, several plans are implemented in the condition of transportation. But these schemes are not well maintained throughout the entire city. This is leading us towards bigger problems like increased dependence upon private sector for transport, travel fares, conditions of roads, increase in number of accidents on roads etc. Among these, the biggest issue that needs a solution today is the poor state of public transport.

The road conditions are severely damaged due to many factors such as laying pipelines, underground cables etc. Taking the development of condition of transportation as the central aim, in this paper a game theoretical approach is used based on [1], for which data has been collected from residents of Salem about the transportation condition that needs to be improved in their localities, to find out which factor has to be developed to the most ultimately so that it becomes a sophisticated area.

Game theory is the study of behaviour in conflict situations. Game theory also includes conflict situations that have almost no relation to games. In order to use the data in game models they have to be properly processed, if the problem has the property of a conflict situation, game theory can be used [5].

1.1 PRELIMINARIES

Definition 1.1: [6] If X is a collection of objects denoted generally x, then a fuzzy set 𝐴̃ in X is defined as a set of ordered pairs 𝐴̃= {𝑥,𝜇𝐴�(𝑥)

𝑥 ∈ 𝑋}. Where 𝜇𝐴�(𝑥) is called the membership function (or grade membership value of x) for

the fuzzy set 𝐴̃. The membership function which maps each element of X to a membership value between 0 and 1.

Definition 1.2: [2] A fuzzy number 𝐴̃= (𝑎𝐿,𝑎𝑈,𝛼₁,𝛼₂) is called a trapezoidal fuzzy number if its membership function meets the following mapping:

f(x) =

⎩ ⎪ ⎨ ⎪

⎧ 𝑥−�𝑎𝐿−𝛼1𝛼1� �𝑎𝐿−𝛼1�≤𝑥≤𝑎𝐿 1 𝑎𝐿_{≤𝑥≤𝑎}𝑈 �𝑎𝑈_{+𝛼2�−𝑥}

𝛼2 𝑎

𝑈 _{≤ 𝑥 ≤(𝑎}𝑢_{+ 𝛼}

2)

0 𝑜𝑡ℎ𝑒𝑟𝑠

2. IMPLEMENTATION OF MATRIX GAMES FOR IMPROVING THE TRANSPORTATION

In this section, the working method of the problem in consideration is given in detail.

Initially, the general problems the residents face in terms of transportation are taken into account. Factors such as conditions of roads, maintenance of roads during adverse conditions, the preference of people towards public/private transport, the expenses of their travel etc are taken into consideration. The aim is to find out the factors that are to be developed in the city. In a city with more than hundreds of areas, different areas require different factors to be developed. So, in order to propose a consolidated outcome, here game theoretical model is used.

The structure of the game is taken as (𝐴,𝐵,𝐿), where A denotes the group of people who have specified the developmental factors, B is the collection of those parameters and L is the payoff matrix of the game. Based on [3], the group of people are represented as 𝐴= {𝑎₁,𝑎₂, … ,𝑎_𝑖}where 𝑎₁ is the group of people who have the notion that all the factors are best in their area, 𝑎₂ is the group of people who have the view that all the parameters specified are moderate(somewhat lesser quality than mentioned by 𝑎₁) in their area and so on. The collection 𝐵= {𝑏₁,𝑏₂, … ,𝑏_𝑗} represents the factors of maintenance, expenses and so on.

If the factors that needs development are not improved then there is a chance that people feel a little mutilated towards the introduction of new schemes on these factors. Without improving the existing flaws, if these factors are given some additional improvisation, the ultimate outcome will be a loss. The matrix of loss for these factors upon the group of residents is given below:

Table-1: Matrix of Losses B

A

𝑏1 . .. 𝑏𝑗

𝑎1 𝐿(𝑎1,𝑏1) . .. . ..

. .. . .. . .. . ..

𝑎𝑖 . .. . .. 𝐿(𝑎𝑖,𝑏𝑗)

Now, a collection of questions are given to the residents and based on their preferences they are grouped as

𝐶= {𝑐1,𝑐2, … ,𝑐𝑘}, where 𝑐1 denotes the residents who prefer the factor 𝑏1 as good, 𝑐2 denotes the residents who prefer

the factor 𝑏₂ as good, and also special emphasis is given to 𝑐_𝑘 which denotes residents who do not have any opinion upon the development. The distribution of probabilities for {𝑐₁,𝑐₂, … ,𝑐_𝑘} based on {𝑎₁,𝑎₂, … ,𝑎_𝑖} is given by:

𝑃�𝑐_𝑖�𝑎_𝑗�,𝑖= 1,2, … . ,𝑘 and 𝑗= 1,2, … . ,𝑛

We now define a mapping 𝑆 ∶ 𝐶 → 𝐵 based on [4] which gives all the possible combinations of the resident's views on the transportation factors. The total number of combinations for the factors 𝐵= {𝑏₁,𝑏₂,… ,𝑏_𝑗} based on the collection

𝐶= {𝑐1,𝑐2, … ,𝑐𝑘} will be 𝑗𝑘 which are given in the collection 𝑆= {𝑠1,𝑠2, … ,𝑠𝑘, where 𝑚= 𝑗𝑘}

Table-2: Selected sample of correspondent factors S

C 𝑠1 𝑠2 𝑠3 𝑠4 𝑠5 . ..

𝑐1 𝑏1 𝑏1 𝑏1 𝑏1 𝑏2 . ..

𝑐2 𝑏2 𝑏2 𝑏2 𝑏1 𝑏2 . ..

. .. . .. . .. . .. . .. . .. . ..

© 2018, IJMA. All Rights Reserved 167 From the matrix of loss of Table 1 and the probability distribution we calculate the functions of risk as follows:

𝑅�𝑎𝑖,𝑠𝑗�= 𝑃�𝑐𝑖�𝑎𝑗�𝐿(𝑎𝑖,𝑠𝑗(𝑋))

If there is no saddle point in the matrix of risk, we use linear programming problem method to solve the optimal solution.

3. ESTABLISHMENT OF THE PROPOSED MODEL FOR THE DATA COLLECTED FROM PEOPLE

Initially the data was collected from the residents about the condition of transportation in their area. Based on their opinions it is found that the factors such as conditions of roads, preference for public/private transport, expenses they bear for travel, frequency of the transportation were mostly having oscillated views among various residents. Hence, the game theoretical model is applied to these strategies to find out which factor among these must be improved still. The collected data is formulated in the following manner:

Let the set of decisions be 𝐵= {𝑏₁,𝑏₂,𝑏₃}, where, 𝑏₁ corresponds to the condition of roads in the city, , 𝑏₂ is the expenses of travel, 𝑏₃ is the frequency of the transportation in the city. 𝐵 is the collection of all these parameters.

Let the people who prefer each of these factors be grouped as 𝐴= {𝑎₁,𝑎₂,𝑎₃} where, 𝑎₁ is the group of people who have best views upon most of the strategies, 𝑎₂ is the group of people who have good views upon all the strategies and 𝑎3 is the group of people who have poor opinion upon all the strategies. We now calculate the matrix of loss. For

example, if we consider the condition of roads in a Kitchipalayam area which is in a bad condition, the condition of that road must be developed first. Without developing the existing condition of roads, if new routes are being laid in the same area, it is doubtful whether that will be welcomed by the residents, which is taken as the loss. Similarly, the loss for other factors such as expenses, frequency of transportation are considered. This matrix of loss 𝐿(𝐴,𝐵) is presented in the following table:

Table-1: Matrix of Losses B

A

𝑏1 𝑏2 𝑏3

𝑎1 5 10 0

𝑎2 0 5 5.

𝑎3 5 10 10

Now, the people who gave their views on the transportation factors are grouped as 𝐶= {𝑐₁,𝑐₂,𝑐₃,𝑐₄} where, 𝑐₁ denotes that the people of the city prefer the factor 𝑏₁ as good, 𝑐₂ denotes that the people of the city prefer the factor 𝑏₂ as good, 𝑐3 denotes that the people of the city prefer the factor 𝑏3 as good, and finally, 𝑐4 denotes the people who have no

preference for any of the factors.

Let the probability distribution of the results {𝑐₁,𝑐₂,𝑐₃,𝑐₄} depending on the initial data {𝑎₁,𝑎₂,𝑎₃} be:

𝑃(𝑐1|𝑎1) = 0.2; 𝑃(𝑐1|𝑎2) = 0.7; 𝑃(𝑐1|𝑎3) = 0.09; 𝑃(𝑐2|𝑎1) = 0.07; 𝑃(𝑐1|𝑎2) = 0.1; 𝑃(𝑐2|𝑎3) = 0.4; 𝑃(𝑐3|𝑎1) = 0.7;

𝑃(𝑐3|𝑎2) = 0.1; 𝑃(𝑐3|𝑎3) = 0.4; 𝑃(𝑐4|𝑎1) = 0.03; 𝑃(𝑐4|𝑎2) = 0.1; 𝑃(𝑐4|𝑎3) = 0.01

To find the optimal strategy, we now introduce a function which maps 𝐶to 𝐵, so that each and every view of the residents'are analysed. Since there are 34 = 81 combinations, instead of analysing all these we consider the trivial combinations of their views for the factors. Let them be denoted as 𝑆= {𝑠₁,𝑠₂,𝑠₃,𝑠₄,𝑠₅,𝑠₆} these strategies are the combination of the factors that are taken into consideration and which are represented in a matrix as follows:

Table-2: Selected sample of correspondent factors S

C 𝑠1 𝑠2 𝑠3 𝑠4 𝑠5 𝑠6

𝑐1 𝑏1 𝑏1 𝑏1 𝑏1 𝑏2 𝑏3

𝑐2 𝑏2 𝑏2 𝑏2 𝑏1 𝑏2 𝑏3

𝑐3 𝑏3 𝑏3 𝑏3 𝑏1 𝑏2 𝑏3

𝑐4 𝑏1 𝑏2 𝑏3 𝑏1 𝑏2 𝑏3

Now, the formula for the function of risk is given by 𝑅�𝑎𝑖,𝑠𝑗�= 𝑃�𝑐𝑖�𝑎𝑗�𝐿(𝑎𝑖,𝑠𝑗(𝑋))

𝑅(𝑎1,𝑠1) = 5.(0.2) + 10.(0.07) + 0.(0.7) + 5.(0.03) = 1.85

𝑅(𝑎2,𝑠1) = 0.(0.7) + 5.(0.1) + 5.(0.1) + 0.(0.1) = 1

𝑅(𝑎3,𝑠1) = 5.(0.09) + 10.(0.4) + 10. (0.4) + 5.(0.01) = 8.5

The function of risk 𝑅(𝐴,𝑆) for the selected 𝑠_𝑖 is given

𝐴 𝑠₁ 𝑠₂ 𝑠₃ 𝑠₄ 𝑠₅ 𝑠₆ 𝑅𝑜𝑤𝑚𝑖𝑛 𝑎1

𝑎2

𝑎3

�1.85 2_{1 1.5 1.5}1.7 5 10 0_{0 5 5}

8.5 8.55 8.55 4.5 9 9�

0 0 4.5 𝐶𝑜𝑙𝑢𝑚𝑛𝑀𝑎𝑥 8.5 8.55 8.55 5 10 9

Since, there is no saddle point, we now proceed to solve this by fuzzy linear programming problem method [4]. Now, the probabilities for the strategies are introduced as 𝑝₁=𝑃(𝑠₁), 𝑝₂=𝑃(𝑠₂), 𝑝₃=𝑃(𝑠₃), 𝑝₄=𝑃(𝑠₄), 𝑝₅=𝑃(𝑠₅), 𝑝6=𝑃(𝑠6).

The fuzzy linear programming problem for the strategies based on the probabilities is given by,

Min 𝑥 = (9,10,13,14)𝑥₁ + (11,13,14,15)𝑥₂ + (10,12,13,14)𝑥₃ + (7,8,10,11)𝑥₄ + (22,23,25,26)𝑥₅ + (11,12,13,15)𝑥₆ subject to the constraints,

1.85 𝑝₁ + 2 𝑝₂ + 1.7 𝑝₃ + 5 𝑝₄ + 10 𝑝₅≤𝑥 𝑝1 + 5 𝑝2 + 1.5 𝑝3 + 5 𝑝5 + 6 𝑝6≤𝑥

8.5 𝑝₁ + 8.55 𝑝₂ + 8.55 𝑝₃ + 4.5 𝑝₄ + 9 𝑝₅ +9 𝑝₆≤𝑥 𝑝1+ 𝑝2 + 𝑝3 + 𝑝4 + 𝑝5 + 𝑝6 = 1

𝑝1 ≥0, 𝑝2 ≥0, 𝑝3 ≥0, 𝑝4 ≥0, 𝑝5 ≥0, 𝑝6 ≥0

Since 𝑥> 0, the above three inequalities are divided by 𝑥 and a new variable 𝑦_𝑖= 𝑝𝑖

𝑥 is introduced.

Hence, we obtain the new fuzzy linear programming problem

Max 𝑧 = (9,10,13,14)𝑦₁ + (11,13,14,15)𝑦₂ + (10,12,13,14)𝑦₃ + (7,8,10,11)𝑦₄ + (22,23,25,26)𝑦₅ + (11,12,13,15)𝑦₆ subject to the constraints,

1.85 𝑦₁ + 2 𝑦₂ + 1.7 𝑦₃ + 5 𝑦₄ + 10 𝑦₅≤1 𝑦1 + 5 𝑦2 + 1.5 𝑦3 + 5 𝑦5 + 6 𝑦6≤1

8.5 𝑦₁ + 8.55 𝑦₂ + 8.55 𝑦₃ + 4.5 𝑦₄ + 9 𝑦₅ +9 𝑦₆≤1 𝑦1 ≥0, 𝑦2 ≥0, 𝑦3 ≥0, 𝑦4 ≥0, 𝑦5 ≥0, 𝑦6 ≥0

Using the fuzzy LPP method, the following solution is obtained. 𝑦1 = 0, 𝑦2 = 0, 𝑦3 = 0, 𝑦4 = 0, 𝑦5 = 0.1, 𝑦6 = 0.01

𝑧1 = 0.11 and hence the value of the game 𝑥 = 9.09

Now, 𝑦_𝑖= 𝑝𝑖

𝑥, 𝑝1 = 0, 𝑝2 = 0, 𝑝3 = 0, 𝑝4 = 0, 𝑝5 = 0.1, 𝑝6 = 0.01

Thus, probability distribution is 𝑃{𝑠5} = 0.90, 𝑃{𝑠6} = 0.09

This means that the final strategy must be determined based on the strategies 𝑠₅ and 𝑠₆. Now, from Table 2, we can see that 𝑠₅ and 𝑠₆ differ in all the entries. Hence the final decision is to choose between 𝑏₂ and 𝑏₃. Now the probability of 𝑠5 is greater than the probability of 𝑠6. Hence, the strategy 𝑏2 is in the best condition. Thus, the conditions of roads

must be improved and the frequency of transportation must be increased in and around the city of Salem, so that the residents welcome all other new plans.

4. CONCLUSION

Game theory plays a major role in decision making process. The strategies are fixed in such a way that each and every conditions are taken into account. In this paper, as a part of aiming to change Salem as a smart city the factors that needs development regarding transportation are taken into account. Based on the residents' views the strategies are framed and the final solution is given. This solution is provided for the situation when there is a dilemma between the selection of factors that requires development.

ACKNOWLEDGEMENT

© 2018, IJMA. All Rights Reserved 169 REFERENCES

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